Yuan-Pin Lee

Quantum Lefschetz hyperplane theorem

Abstract.The mirror theorem is generalized to any smooth projective variety X. That is, a fundamental relation between the Gromov–Witten invariants of X and Gromov–Witten invariants of complete intersections Y in X is established.

Quantum

This work is devoted to the study of the foundations of quantum K-theory, a K-theoretic version of quantum cohomology theory. In particular, it gives a deformation of the ordinary K-ring K(X) of a smooth projective variety X, analogous to the relation between quantum cohomology and ordinary cohomology. This new quantum product also gives a new class of Frobenius manifolds.

K

The main goal of this paper is to introduce a set of conjectures on the relations in the tautological rings. In particular, this framework gives an efficient algorithm to calculate all tautological equations using only finite dimensional linear algebra. Other applications include the proofs of \emph{Wittens conjecture} on the relations between higher spin curves and Gelfand--Dickey hierarchy and \emph{Virasoro conjecture} for target manifolds with conformal semisimple quantum cohomology, both for genus up to two.

-theory, I: Foundations

where C? are connected components of C, C Uf=1C{. By the definition given in [16], the curves in the image of t/ have at most one more connected component. Therefore, the connected components of the image curves would have either smaller genus or the same genus but less marked points than those of the domain curves. It is therefore easy to see that these operators give an algorithm in producing tautological relations inductively. This was explained in Part I. In Part II, the focus moves to Gromov-Witten theory. One of the main themes of the current work is to study the interplay between Gromov-Witten theory and the moduli of curves. The idea of studying and utilizing this interaction is not new. One such example is given by the localization technique in Gromov-Witten theory. The fixed point loci on moduli of stable maps to projective spaces are moduli of curves (or more precisely their quotients by finite groups). Therefore, the known results on moduli of curves can be used to calculate Gromov-Witten invariants.

Invariance of tautological equations I: conjectures and applications

Abstract In this work, we continue our study initiated in [11]. We show that the generating functions of Gromov–Witten invariants with ancestors are invariant under a simple flop, for all genera, after an analytic continuation in the extended Kähler moduli space. The results presented here give the first evidence, and the only one not in the toric category, of the invariance of full Gromov–Witten theory under the K-equivalence (crepant transformation).

Invariance of tautological equations II: Gromov-Witten theory

The celebrated Mirror theorem states that the genus zero part of the A model (quantum cohomology, rational curves counting) of the Fermat quintic threefold is equivalent to the B model (complex deformation, variation of Hodge structure) of its mirror dual orbifold. In this article, we establish a mirror-dual statement. Namely, the B model of the Fermat quintic threefold is shown to be equivalent to the A model of its mirror, and hence establishes the mirror symmetry as a true duality. 14N35; 53D45

Invariance of Gromov-Witten theory under a simple flop

We prove that all monomials of

A mirror theorem for the mirror quintic

\kappa

On the independence of the generators of tautological rings

-classes and

The quantum orbifold cohomology of weighted projective spaces

\psi